In the games the main character always and their rival always start off with their Pokemon Professor giving them a starter pokemon (with the exception to Gold/Silver/Crystal where the rival stole their starter from the professor). in the anime Ash, Gary, 2 unnamed trainers, May and Dawn all got their Pokemon Professor.
But given that there appears to be only 1 Pokemon Professors in each region (Oak = Kanto, Birch = Hoenn, Ivy = Orange Islands, etc.) and we have seen there is so many Pokemon Trainers in the League Qualifiers in the anime it's seems that if every trainer went to the same Pokemon Professor they would be inundated with work looking after everyone's pokemon when they catch more than 6, so much so that Oak shouldn't really have the time to be jet setting around the world like he does for radio shows.
So how does one start off as a Pokemon Trainer if they don't have a Pokemon Professor?